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This time we're going to talk about profit maximization

when we're considering the level of inputs to use.

So this is a really very powerful use

of the production function concept.

Because the production function helps

us to determine the best level of a production input,

such as a fertilizer or a herbicide or water.

So we saw in the last segment about the relationship

between inputs and outputs being captured by a production

function and production functions having

a reasonably characteristic shape.

In this graph you can see two different production functions

for the same crop-- it's wheat in both cases--

but growing on two different soil types.

So it illustrates-- and again, it's nitrogen fertilizer--

so again, it's a relationship between the inputs and outputs.

But you can see that the relationship is a little bit

different depending on the circumstances.

And in this case, it's different between two different soil

types, a sandy soil or a limey soil.

Production's a little bit more responsive, a bit steeper

on a sandy soil because it has less nutrients to start with.

So that previous graph was the relationship

between fertilizer rates and yield.

This graph is the relationship between fertilizer rate

and revenue.

So what I've done to produce this graph

is multiply the numbers in the previous graph

by the price, the sale price, of the wheat

so that you can calculate the amount of revenue that

would be received for each level of fertilizer.

So it has the same shape.

It's just at a different level.

And the axis on the left-hand side now is revenue.

But broadly it's a very similar looking graph.

Next, we need to consider, if we're

trying to determine the optimal level of this input-- nitrogen

fertilizer, in this case-- we need

to worry about how much that's going to cost.

So we're going to plot on this same graph a variable cost

curve.

So the variable cost function is simply

the quantity of fertilizer multiplied by the fertilizer

price.

And naturally enough, that's a straight line.

The more fertilizer we put on, the more it's going to cost us.

And it increases in a linear way like that.

So the left-hand axis now is fertilizer cost.

The bottom axis is, again, nitrogen fertilizer rate.

And now I want to bring those two things together.

The profit is simply the revenue function

minus the cost function.

And we're going to determine the optimal level of the input

by looking for the fertilizer level which has the greatest

difference between revenue and cost.

Where does the revenue exceed the cost

by the greatest amount?

Or where's the biggest gap between the two curves?

So there's the two curves, the two revenue curves,

for the two different soil types.

And the cost function will be the same in each case.

The fertilizer costs the same no matter

which soil type you apply it to.

But you can see the two dotted lines indicate

that the optimal fertilizer rate is different on these two

different soil types.

On the limey soil, which has the higher and less steep response

function, production function, at the top,

you can see that the optimal level of fertilizer

is a bit lower.

Whereas for the sandy soil, which

is a bit more responsive to fertilizer,

the optimal fertilizer rate is understandably a bit higher.

So this is a really helpful concept.

Depending on the relationship between inputs and outputs

and the cost of the fertilizer, you

can calculate which is the optimal fertilizer rate.

And if you eyeball those two different cases,

you can see that the optimal fertilizer rate occurs

where the slope of the cost function

is exactly the same as the slope of the revenue function.

So think about that.

The two slopes are identical.

At those points where the dashed lines are,

if you run your eyes up and down,

you can see that the slope of the revenue function

is the same as the slope of the cost function.

And because the slopes of the curves

are different for the two production functions,

the point where the slopes are equal with the cost function

is also different.

Now in this case, I've just looked

at one of these production functions or revenue

functions, the one for sandy soil.

And I've calculated the difference

between the revenue and the cost.

And that's the new curve in the middle, the purple curve,

which is labeled profit.

So this is just the difference.

So you can find out the optimal level of fertilizer

just by finding the point of this curve, which

is as high as possible, which is the peak of the hill.

And it's the same as we saw in the previous graph.

You can see it's the point where the dashed line is there.

And if we look at the profit curves for both of our soil

types, you can see, again, we've got the-- the peak of the curve

is at two different places.

It's the same as we saw when we looked

at the revenue curve and the cost curve.

Just this time we've only shown the curve

that is the difference between those two curves in each case.

So there's various ways you can identify the optimal herbicide

rate or the optimal fertilizer rate

or the optimal water rate, application rate.

One way is just to judge it from looking at the graph in the way

that we were doing just then.

Another is to use calculus to calculate

a function for the maximum point.

And a third way is to calculate profit numerically

for a range of different input levels.

So just do the maths.

Calculate it.

So for example, if we assume that the wheat cost

is $250 a ton.

And the fertilizer price is $1.90 a kilogram.

And we have a function for the yield

and how it changes in response to fertilize,

f, which is that function down on the bottom there.

Then I can calculate the profit for any particular level

of fertilizer that I'm interested in.

So that's what I've done here.

If you look across the columns, I've

got different fertilizer rates.

I've got the wheat yield.

I've got the revenue, then the cost, and then the profit.

Profit is last column.

So you can see that the-- looking down

the column of the profit figures,

I can identify which of those profit figures

is the highest, the one that's red.

And then look across to the left to see the level of fertilizer

that corresponds to that highest level of profit.

So this is a fairly simple calculation to do.

You could do it in a spreadsheet.

The hardest part is knowing what the function

is for the relationship between fertilizer and yield.

So in summary, the revenue function--

or the production function multiplied by the output price

gives you the revenue function.

And profit is simply the revenue minus the cost.

And we can find the level of an input that maximizes profit

by seeing where the difference between the revenue

and the cost is the greatest.